It is well understood that in communications systems, the shape of a pulse used to transmit information can have an effect of the performance of the system. A goal of pulse shape design is to provide a shape which does not suffer from intersymbol interference (ISI), while at the same time uses as little excess bandwidth as possible beyond the theoretical minimum required to transmit a given amount of information.
Nyquist's historical paper entitled “Certain Topics in Telegraph Transmission Theory,” AIEE Trans., vol 47 pp. 617–644, 1928 introduced a class of ISI-free pulse shapes now commonly referred to as Nyquist pulses. The so-called “raised cosine” pulse is a special case of a Nyquist pulse which has become prevalent in modern systems, for example communications modems, and is the benchmark pulse in modern communications theory. It is noted that for historical reasons, a Gaussian pulse shape is still widely employed in European applications, despite its inferior performance characteristics. The terms “Nyquist pulse” and “raised cosine pulse” have become somewhat synonymous, although more accurately, a “Nyquist pulse” is any ISI-free pulse. The raised cosine pulse with relative excess bandwidth α, is specified by its overall frequency spectrum:
                              S          ⁡                      (            f            )                          =                  {                                                                      1.                                                                      0                    ≤                    f                    ≤                                          B                      ⁡                                              (                                                  1                          -                          α                                                )                                                                                                                                                                                    1                      2                                        ⁢                                          {                                              1                        +                                                  cos                          ⁡                                                      (                                                                                          π                                                                  2                                  ⁢                                  B                                  ⁢                                                                                                                                          ⁢                                  α                                                                                            ⁢                                                              (                                                                  f                                  -                                                                      B                                    ⁡                                                                          (                                                                              1                                        -                                        α                                                                            )                                                                                                                                      )                                                                                      )                                                                                              }                                                                                                                                  B                      ⁡                                              (                                                  1                          -                          α                                                )                                                              ≤                    f                    ≤                                          B                      ⁡                                              (                                                  1                          +                          α                                                )                                                                                                                                          0.                                                                                            B                      ⁡                                              (                                                  1                          +                          α                                                )                                                              ≤                    f                                                                        ⁢                                                  ⁢                                                                                                                              S                        ⁡                                                  (                          f                          )                                                                    =                                              S                        ⁡                                                  (                                                      -                            f                                                    )                                                                                      ⁢                                                                                                                                                      f                    ≤                    0                                                                                                          (        1        )            where B is the bandwidth corresponding to symbol repetition rate T=½B, and its corresponding (scaled) time function is given by 
                                          p            RC                    ⁡                      (            t            )                          =                  sin          ⁢                                          ⁢                      c            ⁡                          (                              t                /                T                            )                                ⁢                                    cos              ⁡                              (                                  2                  ⁢                  π                  ⁢                                                                          ⁢                  α                  ⁢                                                                          ⁢                                      t                    /                    T                                                  )                                                    1              -                              4                ⁢                                  α                  2                                ⁢                                                      t                    2                                    /                                      T                    2                                                                                                          (        2        )            Excess bandwidth is a reference to the allowed bandwidth compared to the theoretical minimum bandwidth required to transmit data at a specified rate (symbol, baud, bit). Practical systems use “excess” bandwidth as real systems are not perfect; timing recovery is hard to do if the excess bandwidth is small. The second generation IS-54 (USA) standard specifies α=0.35, the second generation PDC (Japan) standard specifies α=0.5. Some satellite modems use α=1.0.
The more the excess bandwidth, however, the fewer the number of available channels for a given amount of spectrum. In an ideal world, one would use 0% excess bandwidth. Typically the overall pulse shape is implemented by putting the square root of the spectrum in the transmitter filter with the matched filter in the receiver also having the square root of the pulse spectrum as its frequency response. That is, the pulse is split by taking its square root in frequency and putting half the response in the transmitter and half the response in the receiver. This is known to maximize the signal-to-noise ratio and minimize the average error rate. Thus, the transmitter has a root raised cosine pulse shaping and the receiver matched filter has a root raised cosine shaping.
This pulse shape is characterized by i) certain error rates for different channels, signal-to-noise ratios, and for different symbol timing errors and ii) a certain receiver eye diagram.
Examination of the above equations indicates that the tails of the raised cosine pulse for α>0 decay asymptotically as t−3 as is well known. Attempts to develop new pulse shapes to improve upon the raised cosine pulse have focused on pulse shapes with higher rates of decay, for example with asymptotic decays as t−4, t−5 with the expectation that this higher rate of decay would somehow yield a performance benefit.